Table 1.

HSMC results for the entropy per bond of N-bond SAWs on a square lattice obtained in White and Meirovitch, 2005^a

nf	SA / kB	σ A	SB / KB	$S_{\mathrm{G}}^{\mathrm{B}}∕k_{\mathrm{B}}$	SM / kB	$S_{\mathrm{G}}^{\mathrm{M}}∕k_{\mathrm{B}}$	SD / kB	n
N = 99 SSCAN = 0.987726 (5)
500	0.99294 (2)	0.01030 (3)	0.9826 (1)	0.98243 (6)	0.98775 (5)	0.98769 (4)	0.98773 (5)	250000
5000	0.98826 (2)	0.00324 (3)	0.98722 (5)	0.98722 (3)	0.98774 (3)	0.98774 (2)	0.98774 (3)	25000
50000	0.98777 (2)	0.00101 (3)	0.98767 (4)	0.98767 (2)	0.98772 (2)	0.98772 (2)	0.98772 (3)	2500
S HS	0.98994 (1)	0.00507 (1)	0.9856 (2)	0.9874 (1)	0.9878 (1)	0.9887 (1)	0.98817 (5)	250000
S TI	0.987727 (3)		0.987727 (3)	0.987727 (3)	0.987727 (3)	0.987727 (3)	0.987727 (3)
S series	0.987730 (3)		0.987730 (3)	0.987730 (3)	0.987730 (3)	0.987730 (3)	0.987730 (3)
N = 399 SSCAN = 0.97567 (4)
500	0.98138 (6)	0.00540 (5)	0.9710 (5)	0.9697 (2)	0.9762 (3)	0.9756 (1)	0.9759 (3)	9500
5000	0.97625 (4)	0.00170 (5)	0.9751 (1)	0.97509 (8)	0.97567 (5)	0.97567 (5)	0.97567 (5)	2000
50000	0.97568 (4)	0.00053 (5)	0.97557 (7)	0.97557 (5)	0.97563 (4)	0.97563 (4)	0.97563 (5)	225
S HS	0.98141 (5)	0.00335 (5)	0.9743 (5)	0.9769 (3)	0.9779 (3)	0.9792 (2)	0.9782 (2)	5500
S TI	0.975655 (8)		0.975655 (8)	0.975655 (8)	0.975655 (8)	0.975655 (8)	0.975655 (8)
S series	0.975652(1)		0.975652 (1)	0.975652 (1)	0.975652 (1)	0.975652 (1)	0.975652 (1)

^aFor SAWs, F/k_BT = −S/k_B (equation (8)), therefore, an upper bound for F becomes a lower bound for S and vise versa. The results were obtained from n reconstructions of a straight chain. S^A (equations (15) and (21)) is an upper bound, and σ_A is its fluctuation (equations (17) and (23)). S^B (equations (25) and (26)) and its Gaussian approximation, $S_{\mathrm{G}}^{\mathrm{B}}$ (equation (30)) are lower bounds, and their averages with S^A are denoted S^M (equation (27)) and $S_{\mathrm{G}}^{\mathrm{M}}$ (equation (30)), respectively. S^D (equation (33)) is an exact entropy functional. n_f is related to the number of MC steps per bond. The results for S_TI were obtained by thermodynamic integration, and those for S_scan (equation (13)) by the scanning method (f=5) in Meirovitch, 1985a. The results for S_series were obtained by a series expansion formula (equation (20) of White and Meirovitch, 2005), and those for S_HS (equations (15) and (16)) by the HS method (f=8). The statistical error is defined by parentheses: 1.00(3) = 1.00 ± 0.03.